6th Grade Math Quiz: Find Gcf And Lcm

20 questions · exam conditions

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Find Gcf And LcmQuestion 1 of 20

Three alarm clocks ring at different intervals. Clock A rings every 6 minutes, Clock B rings every 9 minutes, and Clock C rings every 4 minutes. If all three clocks ring together at 3:00 PM, at what time will all three ring together again?

3:18 PM

4:12 PM

3:54 PM

3:36 PM

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6th Grade Math Quiz

6th Grade Math Quiz: Find Gcf And Lcm

Practice Find Gcf And Lcm in 6th Grade Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Find Gcf And Lcm, giving you a quick way to practice the rules, question types, and explanations that matter most for 6th Grade Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

3:18 PM
4:12 PM
3:54 PM
3:36 PM (correct answer)

Explanation: When you see a problem about events happening at regular intervals and need to find when they'll occur together again, you're looking for the Least Common Multiple (LCM). This tells you the smallest amount of time that passes before all cycles repeat simultaneously. To find when all three clocks ring together again, you need the LCM of their intervals: 6 minutes, 9 minutes, and 4 minutes. Start by finding the prime factorization of each number:

6 = 2 × 3
9 = 3²
4 = 2²

For the LCM, take the highest power of each prime factor that appears: 2² × 3² = 4 × 9 = 36. So all three clocks will ring together every 36 minutes. Since they all rang at 3:00 PM, add 36 minutes: 3:00 PM + 36 minutes = 3:36 PM. Looking at the wrong answers: Choice A (3:18 PM) represents 18 minutes after 3:00, which is the LCM of just clocks A and C (6 and 4). Choice B (4:12 PM) is 72 minutes later, which would be when they ring together for the second time after 3:00. Choice C (3:54 PM) doesn't correspond to any meaningful LCM calculation and may result from adding the intervals incorrectly. Study tip: When solving "meeting again" problems, always find the LCM of all the given intervals. Don't confuse this with finding when just two of the items will coincide - you need all of them together.

Question 2

Jake needs to arrange chairs in rows for two different events. For Event A, he has 36 chairs, and for Event B, he has 60 chairs. He wants to use the same number of chairs per row for both events, with no chairs left over. What is the greatest number of chairs he can put in each row?

6 chairs per row for both events
12 chairs per row for both events (correct answer)
18 chairs per row for both events
4 chairs per row for both events

Explanation: We need the GCF of 36 and 60. Since 36 = 2² × 3² and 60 = 2² × 3 × 5, the GCF is 2² × 3 = 12. This means 12 chairs per row is the maximum possible. Choice A (6 chairs) is a common factor but not the greatest, Choice C (18 chairs) doesn't divide 60 evenly (60 ÷ 18 = 3.33...), and Choice D (4 chairs) is a common factor but not the greatest.

Question 3

Two friends jog on a circular track. Sarah completes one lap every 8 minutes, and David completes one lap every 10 minutes. They start together at the starting line. How many laps will Sarah have completed when they meet at the starting line for the third time?

15 laps (correct answer)
12 laps
30 laps
10 laps

Explanation: First find when they meet: LCM(8, 10) = 40 minutes. They meet at the start at 0, 40, 80, and 120 minutes. The third meeting is at 120 minutes. In 120 minutes, Sarah completes 120 ÷ 8 = 15 laps. Choice B (12 laps) is what she completes at the second meeting (80 minutes), Choice C (30 laps) doubles the correct answer, and Choice D (10 laps) is David's lap count at the third meeting.

Question 4

Find the greatest common factor of 54 and 72 by using prime factorization.

Which choice correctly gives the prime factorizations and the GCF?

$54=2\cdot 3^3$ , $72=2^3\cdot 3^2$ , so $\text{GCF}=2\cdot 3^2=18$ (correct answer)
$54=2\cdot 3^3$ , $72=2^2\cdot 3^2$ , so $\text{GCF}=2^2\cdot 3^2=36$
$54=2\cdot 3^3$ , $72=2^3\cdot 3^2$ , so $\text{GCF}=2^3\cdot 3^3=216$
$54=2\cdot 3^2$ , $72=2^3\cdot 3^2$ , so $\text{GCF}=2\cdot 3^2=18$

Explanation: This question tests finding the GCF of 54 and 72 using prime factorization, where GCF is the product of the lowest powers of common primes, helping in simplifying fractions or dividing evenly. The GCF is the largest number dividing both, for example, GCF(36,48)=12; LCM uses highest powers, like LCM(6,8)=24 from 2^33. For 54=23^3 and 72=2^33^2, the GCF=2^13^2=29=18, as it takes the minimum exponents. The correct choice is the one with accurate factorizations and GCF=18. A common error is using highest exponents like for LCM, getting 2^33^3=216, or wrong factorization like 54=2*3^2 (which is 18, not 54). To find GCF with primes: factor each number, take common primes with lowest powers, multiply; for LCM, use highest powers. Applications include reducing ratios; mistakes include incorrect exponents, confusing GCF with LCM, or arithmetic errors in multiplication.

Question 5

Two students start jogging at the same time. One completes a lap every 10 minutes and the other completes a lap every 12 minutes. If they start together, after how many minutes will they both be at the starting line together again? (Find $\text{LCM}(10,12)$ .)

20
22
120
60 (correct answer)

Explanation: This question tests finding the LCM of 10 and 12, the smallest time when joggers meet again at the start, aligning their lap cycles. The LCM is the smallest number both divide into, like LCM(6,8)=24; GCF is the largest dividing both, like GCF(36,48)=12. For example, multiples of 10: {10,20,30,40,50,60,...}, of 12: {12,24,36,48,60,...}, common: {60,120,...}, least=60. They meet after 60 minutes, as smaller like 20 isn't a multiple of 12. A common error is choosing GCF=2, or a non-multiple like 22, or product 120 which isn't least. To find LCM: list multiples, find common, choose smallest; or primes: 10=25, 12=2^23, LCM=2^235=435=60. Applications include cycle alignment; mistakes include not picking least or listing errors.

Question 6

Find the least common multiple of 9 and 12. (Think of the smallest positive number that is a multiple of both.)

18
36 (correct answer)
21
108

Explanation: This question tests finding LCM of 9 and 12, the smallest positive multiple of both. LCM is smallest both divide into, like LCM(6,8)=24 from multiples. For 9 and 12, multiples of 9={9,18,27,36,...}, 12={12,24,36,...}, common start at 36, so LCM=36. The correct answer is 36, choice C. Errors include picking GCF 3, or a larger multiple like 108, or arithmetic like 9+12=21. To find LCM: list multiples, find common, choose least; or prime factors 9=3^2, 12=2^23, LCM=2^23^2=36. Applications: cycles coinciding, like events every 9 and 12 days.

Question 7

Use the distributive property to factor the sum $36+8$ using the greatest common factor. Which expression is correct and fully reduced (the numbers inside the parentheses have no common factor greater than 1)?

$8(4+1)$
$2(18+4)$
$4(9+2)$ (correct answer)
$12(3+1)$

Explanation: This question tests using the distributive property to factor a sum like 36+8 using the GCF, where you factor out the greatest common factor and ensure the quotients inside are coprime, meaning they have no common factor greater than 1. The GCF is the largest number dividing both, like GCF(36,48)=12 from common factors {1,2,3,4,6,12}; LCM is the smallest both divide into, like LCM(6,8)=24. For example, to factor 36+8, GCF(36,8)=4, so 4(36/4 + 8/4)=4(9+2), and 9 and 2 are coprime since their factors are {1,3,9} and {1,2} with only 1 in common. The correct expression is 4(9+2), as it's using the GCF and fully reduced. A common error is factoring with a smaller number like 2(18+4), where 18 and 4 share a factor of 2, or using a non-factor like 8(4+1) but 8 doesn't divide 36 evenly since 36/8=4.5. To factor using GCF: (1) find GCF(a,b), (2) divide both by GCF, (3) write GCF(quotient1 + quotient2), (4) verify quotients are coprime. Applications include simplifying expressions; mistakes include not using the greatest factor, leaving common factors inside, or arithmetic errors like wrong division.

Question 8

Two hallway lights flash on a timer. One flashes every 6 seconds and the other flashes every 8 seconds. If they flash together now, after how many seconds will they flash together again (the LCM of 6 and 8)?

14
24 (correct answer)
56
48

Explanation: This question tests finding the LCM, the least common multiple or smallest number both divide into, like when events coincide again. LCM is the smallest number both divide into, for example, LCM(6,8) from multiples of 6={6,12,18,24,...} and 8={8,16,24,...}, common={24,48,...}, least=24. For this timer problem, list multiples to find the first common one at 24 seconds. The correct answer is 24, choice B, as that's when they flash together next. Errors include picking the product 48, which is a multiple but not the least, or a common factor like 2 instead. To find LCM, list multiples of each, find common multiples, and choose the smallest. Prime factorization helps: for LCM take highest powers of all primes; applications are scheduling like these lights.

Question 9

A coach has 45 orange cones and 30 yellow cones. She wants to set up identical stations using all the cones with none left over. What is the greatest number of stations she can make (the GCF of 45 and 30)?

30
10
15 (correct answer)
5

Explanation: This question tests GCF of 45 and 30 for making equal stations with all cones. GCF is largest dividing both, like GCF(36,48)=12. Factors of 45={1,3,5,9,15,45}, 30={1,2,3,5,6,10,15,30}, common={1,3,5,15}, greatest=15. Correct is 15 stations, choice C, with 45/15=3 orange and 30/15=2 yellow each. Error: picking smaller like 5, not greatest, or LCM 90 too big. Find GCF by listing factors, common, largest; or primes 45=3^25, 30=235, GCF=35=15. Use for equal divisions like here.

Question 10

A science class observes a flashing light every 9 seconds and a different flashing light every 12 seconds. If they flash together at time 0, when is the next time they flash together? (Find $\text{LCM}(9,12)$ .)

18 seconds
36 seconds (correct answer)
21 seconds
108 seconds

Explanation: This question tests finding the LCM, which is the least common multiple or smallest multiple of both 9 and 12, representing the next time lights flash together. The LCM is the smallest both divide into, for example, LCM(9,12): multiples of 9 are 9,18,27,36,... and of 12 are 12,24,36,..., common 36,72,... least 36. In contrast, GCF is largest dividing both, but here LCM for timing. For example, LCM(9,12): list multiples, first common=36. The correct answer is 36 seconds. A common mistake is smaller common like 18 (not multiple of 12) or larger like 108. To find LCM: list multiples, pick smallest common; or prime: 9=3^2, 12=2^23, LCM=2^23^2=36; applies to events coinciding like lights or schedules.

Question 11

Two bells ring on a schedule. One rings every 6 minutes and the other rings every 8 minutes. If they ring together at 12:00, after how many minutes will they ring together again? (Find $\text{LCM}(6,8)$ .)

24 minutes (correct answer)
56 minutes
14 minutes
48 minutes

Explanation: This question tests finding the LCM, the least common multiple, the smallest number that is a multiple of both 6 and 8, useful for when events like bell rings coincide again. The LCM is the smallest number both divide into, for example, LCM(6,8): multiples of 6 are {6,12,18,24,...}, multiples of 8 are {8,16,24,...}, common multiples are {24,48,...}, least is 24; for GCF, it's the largest dividing both, like GCF(36,48)=12. Using the distributive property with GCF, for 36+8 with GCF=4, it's 4(9+2) where 9 and 2 are coprime. Here, they ring together again after LCM(6,8)=24 minutes, as 24 is the smallest common multiple. A common error is choosing a larger multiple like 48, or confusing with GCF which is 2, or adding instead of finding multiples. To find LCM, list multiples of each, find common ones, and choose the smallest; or use prime factorization: 6=23, 8=2^3, LCM=2^33=8*3=24. Applications include scheduling, like when cycles align; mistakes include picking non-multiples like 14 or arithmetic errors in listing.

Question 12

56 minutes
24 minutes (correct answer)
14 minutes
48 minutes

Question 13

A bell rings every 6 minutes and another bell rings every 8 minutes. If they ring at the same time now, after how many minutes will they ring together again? (This is $\text{LCM}(6,8)$ .)

56
24 (correct answer)
14
48

Explanation: This question tests finding the LCM, which is the least common multiple or the smallest number that is a multiple of both 6 and 8, representing when the bells ring together again. The LCM is the smallest number both divide into, for example, LCM(6,8): multiples of 6 are 6,12,18,24,... and multiples of 8 are 8,16,24,..., with common multiples 24,48,... and the least is 24. In contrast, the GCF would be the largest dividing both, but here we need LCM for coinciding events. For example, LCM(6,8) can be found by listing multiples and identifying the smallest common one as 24. The correct answer is 24 minutes. A common mistake is adding like 14 or choosing a larger multiple like 48. To find LCM, list multiples of both and pick the smallest common; alternatively, use prime factorization: 6=23, 8=2^3, so LCM=2^33=24; this applies to scheduling or cycle problems like bells or lights.

Question 14

A teacher has 36 red markers and 48 blue markers. She wants to make identical gift bags with no markers left over. What is the greatest number of gift bags she can make? (This is the GCF of 36 and 48.)

6
12 (correct answer)
18
48

Explanation: This question tests finding the GCF, which is the greatest common factor, the largest number that divides both 36 and 48 without a remainder, useful for dividing items into equal groups like gift bags. The GCF is the largest number that divides both, for example, GCF(36,48): factors of 36 are {1,2,3,4,6,9,12,18,36}, factors of 48 are {1,2,3,4,6,8,12,16,24,48}, common factors are {1,2,3,4,6,12}, so greatest is 12. For LCM, it's the smallest number both divide into, like LCM(6,8): multiples of 6 are {6,12,18,24,...}, multiples of 8 are {8,16,24,...}, common multiples are {24,48,...}, least is 24. In this case, the greatest number of gift bags is the GCF(36,48)=12, so she can make 12 bags, each with 36/12=3 red and 48/12=4 blue markers. A common error is choosing a smaller common factor like 6, or confusing with LCM which is 144, or picking the product 3648. To find GCF, list factors of both numbers, identify the common ones, and choose the largest; alternatively, use prime factorization: 36=2^23^2, 48=2^43, GCF=2^23=4*3=12. Applications include dividing resources equally, like the largest group size for markers; mistakes include not selecting the greatest factor or arithmetic errors in division.

Question 15

A teacher has 36 red pencils and 48 blue pencils. She wants to make identical gift bags with no pencils left over. What is the greatest number of gift bags she can make (the GCF of 36 and 48)?

6
12 (correct answer)
18
24

Explanation: This question tests finding the GCF, which is the greatest common factor or the largest number that divides both without remainder, and relates to making equal groups like gift bags using all items. The GCF is the largest number that divides both, for example, GCF(36,48) from factors of 36={1,2,3,4,6,9,12,18,36} and 48={1,2,3,4,6,8,12,16,24,48}, common={1,2,3,4,6,12}, so greatest=12. For this problem, list factors of 36 and 48 to find common ones, identifying 12 as the greatest, meaning the most bags is 12, with 36/12=3 red and 48/12=4 blue per bag. The correct answer is 12, choice B, as it allows identical bags with no leftovers using the maximum number. A common error is picking a smaller common factor like 6, which works but isn't the greatest, or confusing with LCM like 144 which is too big. To find GCF, list factors of both numbers, identify all common ones, and choose the largest. Applications include dividing items into equal groups, like the largest number of bags here.

Question 16

Two students are making equal-length jump-rope pieces from two ropes: one is 45 ft and the other is 60 ft. They want the longest possible piece length that divides both exactly. What is $\text{GCF}(45,60)$ ?

30
15 (correct answer)
10
5

Explanation: This question tests finding the GCF, which is the greatest common factor or the largest number that divides both 45 and 60 without remainder, representing the longest piece length. The GCF is the largest number dividing both, for example, GCF(45,60): factors of 45 are 1,3,5,9,15,45 and of 60 are 1,2,3,4,5,6,10,12,15,20,30,60, with common 1,3,5,15, greatest 15. In contrast, LCM would be for smallest common multiple, but here it's GCF for dividing lengths. For example, GCF(45,60) lists factors, common are 1,3,5,15, greatest=15. The correct answer is 15. A common mistake is choosing smaller like 5 or 10, or largest like 30 which doesn't divide 45. To find GCF: list factors, pick largest common; or prime: 45=3^25, 60=2^235, GCF=35=15; applies to dividing into equal parts like ropes or tiles.

Question 17

Which choice correctly finds the greatest common factor using prime factorization?

$72=2^3\cdot 3^2$ and $90=2\cdot 3^2\cdot 5$

$\text{GCF}=3^2\cdot 5=45$
$\text{GCF}=2^3\cdot 3^2=72$
$\text{GCF}=2^3\cdot 3^2\cdot 5=360$
$\text{GCF}=2\cdot 3^2=18$ (correct answer)

Explanation: This question tests finding the GCF using prime factorization, where GCF takes the lowest powers of common primes for 72=2^33^2 and 90=23^25. The GCF is largest dividing both, for example, common primes 2 and 3, lowest powers 2^13^2=18. In contrast, LCM uses highest powers, like 2^33^25=360. For example, GCF(72,90): min(

2^3,2^1

)=2^1, min(

3^2,3^2

)=3^2, no 5 in 72, so 29=18. The correct choice is 23^2=18. A common mistake is using highest powers like 72 or including extra like 360 or 45. To find GCF with primes: factor both, take lowest powers of commons; this verifies factorizations and applies to simplifying fractions or grouping.

Question 18

A student says: " $\text{LCM}(6,8)=48$ because $6\times 8=48$ ." Which value is actually $\text{LCM}(6,8)$ (the smallest common multiple)?

12
24 (correct answer)
16
48

Explanation: This question tests identifying the actual LCM of 6 and 8, correcting the student's mistake of using the product instead of the least common multiple. The LCM is the smallest number both divide into, like LCM(6,8)=24 from multiples; GCF is the largest dividing both, like GCF(36,48)=12. For example, multiples of 6: {6,12,18,24,...}, of 8: {8,16,24,...}, common: {24,48,...}, least=24, not 48 which is larger. The actual LCM is 24, the smallest common multiple. A common error is multiplying like 68=48, or choosing non-multiples like 12 (not of 8) or 16 (not of 6). To find LCM: list multiples, find common, choose smallest; or primes: 6=23, 8=2^3, LCM=2^3*3=24. Applications include timing; mistakes include using product or not least.

Question 19

60 (correct answer)
20
22
120

Question 20

$4(9+2)$ (correct answer)
$2(18+4)$
$8(4+1)$
$12(3+1)$